Abstract

The problems of weak and strong invariance of a constant multivalued mapping with respect to the heat conductivity equation with time lag are studied. Sufficient conditions of weak and strong invariance of a given multivalued mapping are obtained.

1. Introduction

1.1. Related Works

There are many theoretical and practical problems in control problems with distributed parameters where known methods do not work to solve them. The typical examples of such problems are conservation of temperature of a volume within admissible bounds and deviation from undesirable states.

Note that the works [1–7] were devoted to the problems of invariance of given sets for the controlled systems. In these works, some results on construction of core of liveness, the maximal weak invariant subset of the given set was obtained for the control system.

In the paper [3], a family of trajectories, which is kept to be within the given set until a certain time (viability), is analytically described for control systems given by differential inclusions. The paper [8] deals with the problem of diffusion process control by boundary control.

However, all the works mentioned above relate to control systems with concentrated parameters. In the papers [9, 10], weak and strong invariance of the given set with respect to a system with distributed parameters were studied. Alimov [11], Albeverio and Alimov [12] studied interesting applied control problems on heat distribution by convectors in a volume.

The present paper deals with the problems of weak and strong invariance of given multivalued mapping for the 3rd heat conductivity boundary value problem with time lag. In the equation of this problem, the control parameter appears on the right hand side. We obtain conditions which can be easily checked to determine the invariance of the given constant multivalued mapping.

1.2. Preliminaries

First of all, we recall some definitions. A bounded region is referred to as the region with piecewise smooth boundary if its boundary can be represented as follows: , where is an open set with respect to the topology on induced by the topology in . Moreover, each is a connected surface of class ; that is, for any point there exists a ball of radius such that the piece of the surface is given by the equation of the form , where and .

Let be a region in with piecewise smooth boundary. We will use the letter to denote the following differential operator [13, 14]:
where the functions satisfy the conditions , and
for all . The inequality (2) is called the condition of uniform ellipticity of the operator defined by (1). The domain of the operator is , which is the space of functions that are twice continuously differentiable in and continuous on .

Define the operator by the equation
where denotes the upward unit normal vector on and is a given positive continuous function defined on .

Definition 1. The number for which the following boundary value problem:
has nonzero solution , is called eigenvalue of this boundary value problem, and the solution is called eigenfunction of the boundary value problem.

Since problem (4) is homogeneous, we assume that
As the operator (1) is self-adjoint, then it has a discrete spectrum [13, 14]; that is,(i)there exist a countable set of eigenvalues of the problem (4) such that
(ii)for each eigenvalue , there are a finite number of eigenfunctions corresponding to it such that , where
(iii)the set of all eigenvalues is complete (closed) in the space ; that is, any function from the space can be uniquely represented in the form
where the equality is understood in the following sense:

The Fourier coefficients in the Fourier series (8) of are defined by formula
Since the operator is self-adjoint, in certain conditions on the functions , , and , the Green formula is written in the form
Using these, we construct the following spaces depending on a parameter. Let be any nonnegative number. Denote

We now define inner product and norm in the spaces . Let
Set
It should be noted that and for all .

Denote by and the spaces of continuous functions and summable quadratically measurable functions defined on with the values in , respectively, where is a positive number.

1.3. Auxiliary Statements

Consider the following heat exchange control problem with lag:
with boundary
and initial conditions
where ; . Here, and are abstract functions whose values at each are unique elements of the space ; is a positive fixed number and is a positive number.

Further, we use the same letter to denote the bounded operator that maps into and is defined by the following formula
where . Clearly, we obtain from this that .

The problem (15)–(17) is understood in the sense of the theory of generalized functions (the theory of distributions) with the values in . We will look for the solution of the problem (15)–(17), which is continuous with respect to and its values belong to one of the spaces , and the initial and boundary conditions are considered as equality of elements of these spaces.

In the control problem (15)–(17), the control function is subjected to either constraint
or
Accordingly, in this paper, we consider both of the problems (15)–(19) and (15)–(17), (20). The control function that satisfies either (19) or (20) is called admissible.

Assume that the problem (15)–(17) has a solution , , at some admissible control . Then we have
It follows from (15) that
Also, by definition of the functions , we obtain
Therefore, in view of (11) we have
Denoting
and using the fact that
we deduce from formulas (22) and (24) that
Since the function (21) must satisfy the initial condition , we have
From this and (27) we see that
Thus, (21) and (29) define the formal solution of the problem (15)–(17) on which can be written as follows:

The following assumption will be needed throughout the paper.

Assumption 2. Let functions of the set satisfy the following conditions:

Proposition 3. For any function , , the following inequality holds:
where are the eigenvalues of the operator arranged in decreasing order, is a positive number, and is the Fourier coefficient of the function .

Proof. Indeed, for , using the Cauchy-Schwartz inequality and the fact that
we obtain

Theorem 4. Let , be an admissible control that satisfies the condition (19), and let Assumption 2 be satisfied. Then there exists a unique solution of the problem (15)–(17) in the space .

Proof. To prove the theorem we use the formal representation (30) of the solution of the problem (15)–(17). By definition of the norm in for , we have
where .We use the Cauchy-Schwartz inequality to obtain the following chain of relations:
Therefore,
Next, we expand the integrand on the right hand side of the last inequality; then we use the Cauchy-Schwartz inequality and after that we use Assumption 2 to obtain
Expanding the square in (36), and then using (39) and Assumption 2, we arrive at
Hence, for each .We now show that the function , , is continuous with respect to the norm of the space . We have
where the numbers and will be chosen depending on any .By (29), We have
Combining this with (41), we obtain
Let be an arbitrary positive number. Since the series in (31) and (39) are convergent, one can choose the number so that . Then, we choose so that . Further, similar to (39) we have
It is clear from these inequalities that we can choose so that . From the estimations for obtained above we conclude according to (43) that , meaning that the function , , is continuous.We now show that the problem (15)–(17) has a unique solution. We assume the contrary, that it has two different solutions , , for the same functions and . Then their difference , , as a solution of corresponding homogeneous problem (15)–(17), can be represented in the form of the Fourier series (21), and the Fourier coefficients , , of the function , , are solutions of the following infinite system of differential equations:
Therefore, from the representation (29) we obtain on . Consequently, , , a contradiction. This proves the theorem.

2. Main results

Definition 5. A set is referred to as the strong invariant set on the time interval with respect to the problem (15)–(19) if for any , , and , , , the inclusion holds true on .

Definition 6. A set is referred to as the weak invariant set on the time interval with respect to the problem (15)–(19) if for any , there exists , , , such that the inclusion holds true on .

Further, strong and weak invariant sets of the form will be investigated, where is a positive number. We will find relations among the parameters , , , and in order to guarantee strong or weak invariance of the set on the time interval with respect to the problem (15)–(19) (or (15)–(17), (20)).

Theorem 7. If , the set is strong invariant on with respect to the problem (15)–(19) if and only if

Proof. Let (46) hold true and let , , , and , , be any functions. Using the same reasoning as in the proof of Theorem 4, we get from (40) that
Then
Hence, meaning that the function is nonincreasing. For this reason, , .The same reasoning can be applied for the time interval to obtain . For any positive number , we can continue in this fashion to obtain
that is, is strong invariant.We now turn to the proof of the second part of the theorem. Let the set be strong invariant. Suppose, contrary to our claim, that the inequality (46) fails to hold. Then, clearly, the function is increasing. Choose the functions as follows:
Then
Since the function is increasing on and , therefore ; that is, , which contradicts the strong invariance of the set . This completes the proof of Theorem 7.

Theorem 8. Let be any positive number. If , the set is weak invariant on with respect to the problem (15)–(19).

Proof. Let . Show that, under condition 9, the set is weak invariant with respect to the problem (15)–(17). Let be an arbitrary function from . Set . Then we obtain from the representation (29) of the solution of the problem (27) that
Here, we used the following inequality:
and Assumption 2. Since the function
satisfies
we have . Then in view of (52) we conclude that , . Repeated application of this reasoning to the intervals , enables us to conclude that , , and this completes the proof.

Definition 9. The set is called weak invariant on with respect to the problem (15)–(17), (20) if for any , , there exists , , such that the inclusion holds for all .

Definition 10. The set is called strong invariant on with respect to the problem (15)–(17), (20) if for any , , and , the inclusion holds for all .

Proposition 11. For any function , , for which , the following inequality holds:
where a positive number.

The proof follows from the following chain of relations
here we used the Cauchy-Schwartz inequality and the relation , , .

Theorem 12. If , then the set is not strong invariant with respect to the problem (15)–(17), (20) on , where is any number.

Proof. Let . To prove the theorem, we use the fact that modulus of the admissible control function can take any big values on small time interval. Consider the function
Note that and for all . Assuming , we have
Due to the term , we obtain that there exists , , such that whenever .We now specify the initial function and admissible control function on as follows:
We observe that
Then by the representation (30) we have
Hence
This implies that
Since and for , then . Therefore which shows that the set is not strong invariant, and the proof of Theorem 12 is complete.

Theorem 13. If , then the set is weak invariant with respect to the problem (15)–(17), (20) on .

M. Tukhtasinov and U. Ibragimov, “Invariant sets with respect to the system with lag,” Reports of Academy of Sciences of Republic of Uzbekistan, vol. 15, article no. 6, pp. 12–15, 2011.View at Google Scholar